Why do apps with phone verification send the user a message, rather than have the user send one to them?

It's quite easy to send an SMS message that appears to come from the phone number of your choice without actually controlling that number. And so sending an SMS from a number doesn't verify your ID in the same way as receiving an SMS to a number.

Since no one has mentioned, sending SMS (by customer) does cost money, atleast in developing countries. Besides the validation server can be in a different country. Personally, I won't want to send a costly SMS to US from Japan. Since server sends SMS through 3rd party SMS providers, they don't have to face that much cost per SMS.